11
votes
Formula for volume of a convex polytope
Here is one paper, whose introduction will lead you to others:
Lasserre, Jean B., and Eduardo S. Zeron. "A new algorithm for the volume of a convex polytope." arXiv math/0106168 (2001).
If you ...
- 151k
6
votes
Is there a volume conjecture for closed 3-manifolds?
There's another volume conjecture formulated by Chen and Yang for Turaev-Viro invariants of closed manifolds. They present some evidence for the conjecture in the paper. In a second paper, Yang and ...
- 68.8k
5
votes
Formula for volume of a convex polytope
The volume of the resulting convex polytope is given by the leading coefficient of the Ehrhart (quasi)polynomial if all matrices in your half-space description are rational (that is, have only ...
- 984
4
votes
Accepted
Maximal area/volume of (d-1)-dimensional object in d-dimensional hypercube
The maximal area of a $d-1$ dimensional slice through a $d$-dimensional hypercube is $\sqrt 2$ in any dimension, this was proven by K.M. Ball, Cube slicing in $\mathbb{R}^n$ (1986).
- 188k
4
votes
Accepted
An alternative to Cayley Menger determinant for calculating simplex volume
I'm not sure what is it that you call the Caley-Menger determinant, but its Wikipedia page seems to define it by a formula generalizing yours to arbitrary dimension.
This is arguably much better known ...
- 8,992
3
votes
Accepted
What is the smallest area of a central section of the unit hypercube?
As in the linked paper by K. Ball, it is more convenient to deal with the cube $Q:=[-1/2,1/2]^n$ instead of $\mathcal U$.
In Ball's paper, $f(r)$ is defined, for real $r$, as the volume of the ...
- 128k
3
votes
What is the smallest area of a central section of the unit hypercube?
While the proof in Ball's paper Cube Slicing in $\mathbb{R}^n$ uses probability theory, the result is deterministic. It is proved that the function $f(t) = |(H+ta)\cap Q|$ (where $H$ is the hyperplane ...
- 12.7k
2
votes
Number of distinct points in an n-dimensional tetrahedron
You wish to count solutions to $p_1+\dotsb + p_n \leq D$,
under the condition that $0 \leq p_i<p_{i+1}$.
This is the same as counting solutions to
$$
(p_1+1-1) + (p_2+1-2) + \dotsb + (p_n+1-n) \...
- 15.8k
1
vote
Accepted
Deflating a tetrahedron to a $K_4$ graph with equal changes to sidelengths
Yes, this is true. The main point is that the "first thing that goes wrong" cannot be two vertices coming together.
Let $a_0$, $b_0$, $c_0$, $d_0$, $e_0$, $f_0$ denote the edge lengths of ...
- 25.2k
1
vote
Volume satisfying inequality constraints (simplex subset)
Unfortunately I haven't been able to find any analytic approach for computing the volume. However, there is a simple method for detecting and eliminating superfluous constraints from Piepel (1983) and ...
- 111
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